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ception of algebra began with the British 'reformers*, Peacock,
Herschel, De Morgan, Babbage, Gregory, and Boole. What was
a somewhat heretical novelty when Peacock published his
Treatise on Algebra in 1830 is to-day a commonplace in any
competently written school book. Once and for all Peacock
broke away from the superstitition that the #, y, z, .., in such
relations as x -f y = y -f # #2/ = yx, x(y + z) = xy -f xz, and
so on, as we find them in elementary algebra, necessarily
'represent numbers'; they do not, and that is one of the most
important things about algebra and the source of its power in
applications. The x, y, z, ... are merely arbitrary marks, com-
bined according to certain operations, one of which is symbo-
lized as -f, another by x (or simply as xy instead of x x y),
in accordance with postulates laid down at the beginning, like
the specimens x -f- y = y -f &, etc., above.
Without this realization that algebra is of itself nothing more
than an abstract system, algebra might still have been stuck
East in the arithmetical mud of the eighteenth century, unable
to move forward to its modern and extremely useful variants
under the direction of Hamilton. We need only note here that
this renovation of algebra gave Boole his first opportunity to do
fine work appreciated by his contemporaries. Striking out on
his own initiative he separated the symbols of mathematical
operations from the things upon which they operate and pro-
ceeded to investigate these operations on their own account.
How did they combine? Were they too subject to some sort of
symbolic algebra? He found that they were. His work in this
direction is extremely interesting, but it is overshadowed by
the contribution which is peculiarly his own, the creation of a
simple, workable system of symbolic or mathematical logic.
To introduce Boole's splendid invention properly we must
digress slightly and recall a famous row of the first hah* of the
nineteenth century, which raised a devil of a din in its own day
but which is now almost forgotten except by historians of
pathological philosophy. We mentioned Hamilton a moment
ago. There were two Hamiltons of public fame at this time, one
the Irish mathematician Sir William Rowan Hamilton (1805-
65), the other the Scotch philosopher Sir William Hamilton